Residents at Connors House Residential Care Home got into the festive spirit and enjoyed an afternoon of singing and dancing!
Whilst the weather outside was frightful, the residents enjoyed a fun filled afternoon of live music that they could all sing along to! There was a variety of music from the band with their renditions of Christmas carols, popular music and of course some classic golden oldies. Some of residents even took to their very own dance floor.
John Hill, whose wife visits Connors House on Fridays for day care, manages the band and long with his compatriot's, they very kindly offer their services for free, coming in a few times a year to liven up the lounge! The residents really look forward to the band coming and always give them a very warm reception.
An afternoon with some furry friends was on the cards for residents of Connors House Residential Care Home in Canterbury last week, when they called in to a cat café.
Storm Gareth didn't stop residents of Connors House Residential Care Home in Canterbury having some fun, as they spent a blustery morning baking some tasty treats.
Residents of Connors House Residential Care Home were treated to a special romantic themed afternoon tea party on Valentine's Day.
A community volunteer project between Canterbury Christ Church University and Connors House Residential Care Home is helping to develop relationships between the old and young.
The Lord Mayor of Canterbury, Councillor Colin Spooner, visited residents at Connors House Residential Care Home last week for a New Year visit.
Residents of Connors House Residential Care Home got into the festive spirit and celebrated Christmas at their annual party.
Christmas spirit was in the air at Connors House Residential Care Home this week, when the Emmanuel Church dropped in to sing some festive carols for the residents.
Staff of Connors House Residential Care Home in Canterbury treated residents to a day of Christmas shopping in the city centre and a festive pub lunch.
Residents of Connors House Residential Home in Canterbury hosted a ghoulish get together to celebrate Halloween on Wednesday 31st October.
The strum of sweet music filled the corridors of Connors House Residential Care Home yesterday, when local musician, Jon, dropped in to entertain and inspire the residents.
As the summer draws to a close, residents of Connors House Residential Care Home made the most of the final warm days and headed to Whitstable for a day trip.
As the heatwave continues to bake across the UK, residents of Connors House Residential Care Home did a spot of baking of their own, in the kitchen!
Connors House Care Home in Canterbury hosted its annual summer fete at the weekend, raising an excellent £300 for the residents' amenities fund.
Residents of Connors House Residential Care Home in Canterbury put their gardening skills to good use and decided to grow their own vegetables.
Staff of Connors House Residential Care Home in Canterbury made sure it was a Mother's Day to remember for its residents and their families.
Romance was in the air at Connors House Residential Care Home on Valentine's Day, when residents gathered together for a special Valentine's themed afternoon tea.
Residents of Connors House Care Home in Canterbury enjoyed some flipping good fun on Shrove Tuesday, getting into the spirit of Valentine's Day too!
Residents of Connors House Residential Care Home welcomed visitors for a day of Christmas cheer and singing carols together.
Residents at Connors House Residential Care Home enjoyed an afternoon of entertainment and sang a song or two.
Kitchen staff at Connors House Residential Care Home got into the spirit of Christmas by baking an array of festive delicacies, to raise money for the residents of the home.
Getting into the spooky spirit of Halloween, residents of Connors House Residential Care Home spent a morning carving pumpkins, ahead of fright night.
Residents of Connors House in Canterbury were invited to a magical land this weekend, when Snow White visited the Craddock Road residential care home.
Connors House Residential Care Home hosted its annual summer fete on Saturday 9th September, raising £250 for the residents of the home.
Plenty of tasty tea and cake was on the menu this week for Connors House Residential Care Home residents, as they enjoyed an alfresco afternoon tea party.
Residents of Connors House Residential Care Home pulled on their gardening gloves for an alfresco afternoon sprucing up the outside areas.
Surrounded by seagulls, blue skies and salty sea, residents of the Craddock Road residential care home were transported to a reminiscent seaside setting.
From humble beginnings in Kingswood Road, Gillingham, The Abbeyfield Kent Society has grown to become the biggest independent Abbeyfield in the world. In May, it celebrated its 50th anniversary.
With summer well and truly on its way, residents of Connors House Residential Care Home opted to enjoy the heatwave and take their afternoon tea outside.
Whilst many of us might be thrilled at the thought of having our lunch prepared for us each day, residents at Connors House in Canterbury, decided they'd like to cook for themselves for a change!
Residents at Connors House Care Home in Canterbury enjoyed an Easter themed egg-xtravaganza of activities and events over the weekend.
Staff at Connors House Care Home in Canterbury hosted a special Valentine's themed celebration for the residents of the home on Tuesday 14th February.
Residents of Connors House Care Home in Canterbury grabbed a slice of the action on National Pizza Day, and spent a morning making their own tasty pizzas from scratch.
Sheriff of Canterbury, Councillor Rosemary Doyle, made a festive visit to meet the residents and staff at Connors House Care Home in Canterbury.
Residents and their loved ones indulged in a special night of fine dining at Connors House Care Home in Canterbury, when staff treated them to a restaurant themed evening.
After years of being unable to write anything at all, Pat Sharpe, a resident at Connors House in Canterbury, has signed her first birthday card for several years.
Residents of Connors House Care Home in Canterbury spent a chilly autumn afternoon releasing their inner artists and getting stuck in to their scrapbooks.
The smell of freshly baked treats filled the corridors of the Canterbury care home, as the residents turned their hand to making a delicious batch of pasties.
Two residents and keen artists from separate Abbeyfield Kent care homes spent an afternoon meeting one another and discussing their love of painting over afternoon tea.
Staff and residents at Connors House Residential Home in Canterbury have embraced a week of activities to raise awareness and commemorate Dementia Awareness Week.
Residents, their relatives and staff at Connors House Residential Home were joined by guests, for the official opening of their revamped courtyard.
A number of sons and daughters joined their mothers at Connors House Residential Home on Mother's Day, where they celebrated with a special afternoon tea.


text:
<?php

namespace Sylius\Component\Cart;

final class SyliusCartEvents
{
    const CART_CHANGE            = 'sylius.cart_change';
    const CART_INITIALIZE        = 'sylius.cart.initialize';
    const CART_ABANDON           = 'sylius.cart.abandon';

    const CART_CLEAR_INITIALIZE  = 'sylius.cart_clear.initialize';
    const CART_CLEAR_COMPLETED   = 'sylius.cart_clear.completed';

    const CART_SAVE_INITIALIZE   = 'sylius.cart_save.initialize';
    const CART_SAVE_COMPLETED    = 'sylius.cart_save.completed';

    const ITEM_ADD_INITIALIZE    = 'sylius.cart_item.add.initialize';
    const ITEM_ADD_COMPLETED     = 'sylius.cart_item.add.completed';
    const ITEM_ADD_ERROR         = 'sylius.cart_item.add.error';

    const ITEM_REMOVE_INITIALIZE = 'sylius.cart_item.remove.initialize';
    const ITEM_REMOVE_COMPLETED  = 'sylius.cart_item.remove.completed';
    const ITEM_REMOVE_ERROR      = 'sylius.cart_item.remove.error';
}

redpajama_set_name: RedPajamaGithub


text:
The Area News' trusted source for property
SportLocal Sport
sport, local-sport, football wagga, finals, leonard cup, structure, hanwood, tolland, grand final, tony dobbin
Football Wagga have agreed to review their finals structures for next season but explained it is too late for any changes to be made before the weekend. The decision comes after a number of clubs contacted Football Wagga about finals structures this season, where the make up of Leonard Cup finals series particularly has come under fire. Hanwood, who finished minor premiers, will take on second-placed Tolland in a sudden-death preliminary final on Sunday, while the third and fourth-placed teams, Cootamundra and Junee, meet in the other grand final qualifier. The structure, a top five finals series played over three weeks, does not allow for the top two teams to meet in the grand final, unless Tolland had lost to Cootamundra last weekend. Football Wagga president Tony Dobbin conceded it is something that will be looked at for next year. "We'd agree that we need to look at it, the clubs have come to us and said look at it, and that's certainly what will happen for next year," Dobbin said. "We have committed to the clubs that we will look at it. The best thing for us would be not to have a five team finals, or a five team finals series but over four weeks. "It has been recognised that there is an issue with one v two and we will try and look at it. A couple of clubs have asked us to look at it and we have committed to looking at it." Dobbin said it was too late to change the draw for this weekend. "Yeah, because once it is in the competition rules, we can't, we have the rules in place," he said. "The other aspect that has been raised is that clubs, whether it be a four team, five team or six team competition, they know what it is and the outcome of one, two, three, four, five is how they've played during the year. "We had a very shortened season, compared to our normal season so the issues that we will look at in the future is do we ever have a five team, three week finals series, or we do it over four weeks, or go with the four teams, or go to six teams. All those things will be in the mix." Dobbin said it was not until finals hit that the issue was realised. "I'll be honest, the five team structure has never really been used before and it's not until quite literally we got into it that people realised the way it was going to work," he said. "It was not really realised that team number two, if they won, would meet team number one in the second round, not the last round. "It's a recognition now that yes we would look at it again if we had a five team, three week finals series again. Its the first time we've ever had it, in my understanding, and it's not until you actually get to the draw and think what's going on here." Dobbin said the plan was to return to a full 18-round season next year. READ MORE
/images/transform/v1/crop/frm/8tYDWUpBiaA8SfdG6xkddz/2abe54b3-b2d4-4b7d-8cab-d9d95f60f725.jpg/r4_88_2454_1472_w1200_h678_fmax.jpg
Football Wagga agree to review finals structures after criticism of Leonard Cup series
Matt Malone
FINALS TIME: Tolland's Joanna Balcombe runs away from Cootamundra's Emily Doohan in the Leonard Cup final at Rawlings Park last Sunday. Picture: Les Smith
Football Wagga have agreed to review their finals structures for next season but explained it is too late for any changes to be made before the weekend.
The decision comes after a number of clubs contacted Football Wagga about finals structures this season, where the make up of Leonard Cup finals series particularly has come under fire.
Hanwood, who finished minor premiers, will take on second-placed Tolland in a sudden-death preliminary final on Sunday, while the third and fourth-placed teams, Cootamundra and Junee, meet in the other grand final qualifier.
The structure, a top five finals series played over three weeks, does not allow for the top two teams to meet in the grand final, unless Tolland had lost to Cootamundra last weekend.
Football Wagga president Tony Dobbin conceded it is something that will be looked at for next year.
"We'd agree that we need to look at it, the clubs have come to us and said look at it, and that's certainly what will happen for next year," Dobbin said.
"We have committed to the clubs that we will look at it. The best thing for us would be not to have a five team finals, or a five team finals series but over four weeks.
"It has been recognised that there is an issue with one v two and we will try and look at it. A couple of clubs have asked us to look at it and we have committed to looking at it."
Dobbin said it was too late to change the draw for this weekend.
"Yeah, because once it is in the competition rules, we can't, we have the rules in place," he said.
"The other aspect that has been raised is that clubs, whether it be a four team, five team or six team competition, they know what it is and the outcome of one, two, three, four, five is how they've played during the year.
"We had a very shortened season, compared to our normal season so the issues that we will look at in the future is do we ever have a five team, three week finals series, or we do it over four weeks, or go with the four teams, or go to six teams. All those things will be in the mix."
Dobbin said it was not until finals hit that the issue was realised.
"I'll be honest, the five team structure has never really been used before and it's not until quite literally we got into it that people realised the way it was going to work," he said.
"It was not really realised that team number two, if they won, would meet team number one in the second round, not the last round.
"It's a recognition now that yes we would look at it again if we had a five team, three week finals series again. Its the first time we've ever had it, in my understanding, and it's not until you actually get to the draw and think what's going on here."
Dobbin said the plan was to return to a full 18-round season next year.
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This story Football Wagga to review finals structures first appeared on The Daily Advertiser.
Discuss "Football Wagga to review finals structures"
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redpajama_set_name: RedPajamaCommonCrawl


My family is obsessed with this granola just as much as I am too. It's adapted from one of my all time favorite bloggers, Minimalist Baker, who has some of the very very best plant based recipes, including this one.
1 medium to large ripe banana, mashed (~1/2 cup) ...the spottier and browner the better for maximum heavenliness.
Mix the oats, cinnamon, sugar, salt, flaxseed and nuts together in a large bowl.
In a small saucepan over medium-low heat, warm and melt the coconut oil, maple syrup (or agave or honey), vanilla extract, and maple extract. Once liquidy, remove from heat and whisk in the mashed banana until well combined. Pour over the dry ingredients and mix well.
Spread the mixture evenly onto one or two baking sheets (making sure it doesn't get crowded) and bake for 23-28 minutes or until golden brown. The coconut oil will help it crisp up well, but be sure to watch it carefully as it can brown quickly depending on your oven.
If you don't toss the granola while baking, it will make get clumpy, which I personally love. But if you want a more crumbly granola, toss/stir a bit at the halfway point to break up the clumps.
Once the granola is visibly browned and golden on the edges, remove from the oven and toss just a bit to let the heat escape. Cool completely on the baking sheet or in a heat-safe bowl. It will get really crunchy when it cools. Store in a container or jar with an air-tight seal – it should keep for a couple weeks.


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Two premier Stillness and Echoes concerts - live solo piano music presented with specially-produced short films and moving images - by composer/pianist Robert Bruce. These programs feature all-Canadian content to commemorate Canada's sesquicentennial.
An hour-long concert featuring original neoclassical music with imagery of Ontario nature, built heritage, and artworks.


text:
Q: ember component increment/decrement value I am stuck with increment and decrement a value in a component js. I got this in my template:
<div class="circleBase" {{action 'upHeat' device.status}}> <p>  {{heat}} C </p> </div>

For now I just want to increment the heat value when I am clicking on the div. My component looks like this:
    init() {
        console.log('init');
        this._super(...arguments);
        this.errors = [];
      },
      didRender() {
        this.setProperties({
          heat: 0
        });
      },
    actions: {
        upHeat: function(a){
            this.set('heat', this.heat +1)
            //or
            this.setProperties({
                heat: this.heat +1
              });
           console.log(heat);
        }
      }

This is not working. Each time I click on my div the heat-value is incremented but not saved. My template still shows 0 as value.

A: You have reset the value in didRender().
Therefore the hook will trigger a render which will reset the value to 0.
Use console.log(something) in didRender to see that the didRender hook is invoked whenever the div is clicked.
I have created a twiddle with your requirement. Please have a look.
https://ember-twiddle.com/5ef763cc77aec803f9d62ae85a15dbc1

A: You should never read a value like this.heat in Ember, use this.get('heat') instead. So this would would for you:
this.set('heat', this.get('heat') + 1);

There is another example if you just want to increment a property though, you can use this.incrementProperty like this.
this.incrementProperty('heat');

You can read about in the API docs.

redpajama_set_name: RedPajamaStackExchange


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Rycerze Kolumba w całych Stanach Zjednoczonych utrzymują ostatnio bliskie i regularne kontakty z lożami masońskimi.


text:
The Doors, Morrison Hotel, Los Angeles, CA, 1969
11x14 $600 16x20 $2,500 20x30 $5,500 30x40 $6,500
Unframed $0 Framed $250
Price (excluding sales tax)
© Henry Diltz, 1969
"This was a transient hotel in Downtown LA on Hope Street. The Doors didn't have permission to take this picture, so when the lobby was empty, they ran in quickly and sat behind the window. One roll of film shot before we would get asked to leave." -- Henry Diltz. This became the Morrison Hotel album cover.
The Doors, Morrison Hotel, Los Angeles, CA, 1969 by Henry Diltz
Color: Color Type: Archival Digital Print Edition: Limited Edition (11x14 Open Edition) Signed: Signed
Morrison Hotel Gallery Prints
Prints are generally made to order and delivery usually takes 4 to 6 weeks.
All frames are of high quality wood. VIEW SAMPLE
For information on the technical specifications of our photographic prints, please contact a sales associate at any of our galleries.
Advice on BUYING PHOTOGRAPHIC PRINTS. Please note: if you are located outside of the U.S. you will be responsible for paying any duties and/or taxes imposed by your country before the photographs can be delivered.
If picked up in or shipped to New York, sales tax will be charged at 8.875%. If picked up in Los Angeles, CA, sales tax will be charged at 9.50%. If picked up in or shipped to California, sales tax will be charged at 9.50%. There is no sales tax otherwise. Please note: if you are located outside of the U.S. you will be responsible for paying any duties and/or taxes imposed by your country before the photographs can be delivered.
Related Photographs
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© Joel Brodsky, 1967
The Doors, Hard Rock Cafe, Los Angeles, CA, 1969
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Jim Morrison, Los Angeles, CA 1969
The Doors, Venice Pier, CA 1969
Jim Morrison Triptych, Hollywood Bowl, Los Angeles, CA 1968
Jim Morrison, The Doors, with model Donna Mitchell, 1967
© Condé Nast Collection, 1967
The Doors, Los Angeles, CA, 1968
© Art Kane, 1968
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The Doors, Studio
© Guy Webster, Date Unknown
The Doors, Greenhouse
Original Photographs Selected As Album Covers - Available as Fine Art Photographs
Morrison Hotel Gallery was founded on, and inspired by the photographs Henry Diltz took of the Doors that became their 5th studio album, Morrison Hotel....
Best Rock Photos of the 1960s
Morrison Hotel Gallery is excited to present this gallery of classic rock band photos from the 60s! The decade that brought us some of the most unforgettable...
Rock Music Photography
Rock music is a broad term for a genre that has developed from blues and soul based rock 'n' roll in the early '50s to pop rock, the British invasion,...
Morrison Hotel Gallery Concierge Service
Morrison Hotel Gallery's website has a robust search function, searching thousands of images when you type in an individual band or artist name.
We also offer our Concierge Service so that we can find more specific requests.
List three of your favorite music artists, celebrities or bands you'd like to see in fine art prints.
Be as specific as possible to receive best response from our concierge.
Please give us any additional specifics to help us find images you would like as fine art prints.
Select Genre Activist-Politic Spirit Social Americana Bluegrass Blues Cars Classical Comedy Country Electronic Folk Glam Grunge Hip Hop Hollywood Celebrity Jazz Latin Metal Pop Punk Rap Reggae Rock Soul Sports Select Time Period 1920's 1930's 1940's 1950's 1960's 1970's 1980's 1990's 2000's Monterey Pop Woodstock Select Price Range $200 - $800 $800 - $1000 $1,000 - $3,000 $3,000 - $10,000 $10,000+ SUBMIT


Welcome to my stop on the blog tour for One More Lie by Amy Lloyd, I'm delighted to be sharing my review with you today.
The lure to this novel is the same as the lure to Lloyd's previous novel, The Innocent Wife, and that's the 'ripped from the headlines' nature of the story. A childhood spent in prison – the very idea of children committing a serious crime creates a warped curiosity in us because it's so against the status quo. And what Lloyd does in this novel is feed into that curiosity. We want to know what Charlotte did, and why, how; we want to like her as an adult so we can forgive her actions as a child.
This story had me hooked, and the short chapters served their purpose, to create suspense, to build tension, to feed into my anxiety that everything is going to go wrong for Charlotte again! There's this real sense of awkwardness around Charlotte's character, this sense of not knowing how to 'just be' – and this was a great reflection of how her previous experiences shaped her, and this gave the novel its realism. The ending did seem of a more entertaining nature than the rest of the novel but it worked, and I'd say this is a solid novel for fans of psychological suspense.
One of the things I really liked about this novel was the strength of the narrative voices. You really got a sense of Charlottes thoughts and feelings, and when the narrative voice changed to Sean, it was such a different 'voice', it really felt like two people telling me this story, offering up information that was their truth.
One More Lie is a fantastic exploration of unhealthy relationships and unhealthy attachments, not just between friends but it explores a few of the different relationships people have in their lives, familial and otherwise. And it was this exploration of relationships that was so interesting to me – examining who holds the power in these relationships, and the influence and effect people can have on each other.
If you enjoyed Lloyd's previous novel, you'll enjoy this one too! The realism of the characters and their fascinating natures, the crime and the lies, I definitely recommend this one.
The characterisation is just brilliant! Great review!
Yes, so so brilliant! Thank you!
I find unhealthy relationships fascinating! A wonderful review.
Me too, I love anything with a strong psychology focus!
Outstanding review, Janel💜 Definitely will check this one out.


text:
\section{Introduction}

If primordial Black Holes (BH) \,\cite{ZelNov:1967, Hawking:1971ei, Carr:1974nx} were generated in the early Universe,
they would have emitted, via their evaporation mechanism \cite{Hawking:1974sw}, not only the Standard Model  (SM) particles,
but also all existing particles beyond the SM with mass below their Hawking temperature.
It was soon proposed that such particles might be responsible for the excess of baryons over anti-baryons \cite{Hawking:1974rv, Zeldovich:1976vw},
that they might account for some or all of the Dark Matter (DM) we observe today \cite{Fujita:2014hha, Lennon:2017tqq, Morrison:2018xla, Masina:2020xhk}, and that they might even provide a contribution to Dark Radiation (DR) \cite{Lennon:2017tqq, Hooper:2019gtx, Lunardini:2019zob, Hooper:2020evu}.

The masses of the primordial BHs could be in the broad range $10^{-5}-10^9$ g, {\it i.e.} from the Planck mass up to the mass allowing for evaporation before the nucleosynthesis epoch.
Apart from the case of gravitino production \cite{Khlopov:2004tn, Khlopov:2008qy}, the primordial BH density at formation for the range $10^{-5}-10^9$ g is at present unconstrained, as reviewed \emph{e.g.} in ref.\,\cite{Carr:2020gox}.
There however upper bounds\,\cite{Papanikolaou:2020qtd, Domenech:2020ssp} on the fraction of the universe collapsed into primordial BHs  from possible backreaction gravitational waves.
Depending on the fraction of primordial BHs at formation with respect to radiation, $\beta$, there is the possibility that the Universe was radiation or BH dominated at the evanescence of the BHs \cite{Barrow:1990he, Baumann:2007yr, Fujita:2014hha}: this situation is referred to as radiation or BH domination, respectively.

Much work has been done in the past, and also recently, in the case of non-rotating, {\it i.e.} Schwarzschild BHs, for both DM and DR.
It is natural to ask what changes in the case of rotating, {\it i.e.} Kerr BHs \cite{Kerr:1963ud, Page:1976ki}: this is the goal of the present work.
To motivate the interest in the extension to the Kerr case, let us first summarize the results so far obtained for DM and DR in the Schwarzschild case.

As for DM, Fujita \emph{et al.} \cite{Fujita:2014hha}, assuming BHs domination,
found that a significant contribution could come from stable particles that are either superheavy or light, that is with masses in the MeV range.
In the light case, DM candidates would be warm, while in the superheavy case they would be cold.
Exploiting the warm DM velocity constraints available at that time \cite{Viel:2005qj}, ref.\,\cite{Fujita:2014hha} discussed the lower limits on the mass of the light DM candidates, using an order-of-magnitude argument (for an up-to-date presentation, see \cite{Masina:2020xhk}).
More sophisticated analysis were done in refs. \cite{Lennon:2017tqq, Baldes:2020nuv}.
Ref. \cite{Auffinger:2020afu} presents a complete study on the viability of warm DM candidates from the evaporation of primordial BHs:
it was found that, assuming BH domination, the scenario of warm DM is excluded for all spins and for all BH masses in the range $10^{-5}-10^9$ g;
for radiation domination, upper limits on $\beta$ (or, equivalently, on the warm DM mass) were derived for the various DM spins.

It is natural to ask what happens to warm DM in the Kerr case \cite{Kerr:1963ud, Page:1976ki}.
Since Kerr BHs have shorter lifetimes, one might expect that the mean velocity of DM gets reduced, and the tension with structure formation alleviated.
This is what we study in the next sections, finding that the tension with structure formation is practically unchanged for DM particles with spins $s=0,1/2,1$, while it is even exacerbated for $s=2$ increasing the value of the Kerr BH spin.

As for DR, in the Schwarzschild case, Hooper {\it et al.}\,\cite{Hooper:2019gtx} pointed out that, for BH domination, the contribution to the effective number of relativistic degrees of freedom, $\Delta N_{eff}$, might be at hand of future observations, for DR particles with spins $s=0,1/2,1$, but not for $s=2$.
The effect of primordial BHs merging was recently reconsidered in ref.\,\cite{Hooper:2020evu}, showing that a population of Kerr BHs might be formed and that their evaporation would produce a significant fraction of hot (high-energy) gravitons: exploiting previous results from \cite{Fishbach:2017dwv},
ref.\,\cite{Hooper:2020evu} finds an increase in $\Delta N_{eff}$ at a potentially observable level, even for non extremal BH spin values  (such as $a_*=0.7$), for BHs evaporating just before nucleosynthesis.

Here we reconsider the calculation of $\Delta N_{eff}$ in the Kerr case, for any value of the BH spin and independently of the mechanism responsible for the rotation. In order to numerically account for the greybody factors associated to the different spins, we use the recently developed and publicly available code \texttt{BlackHawk} \cite{Arbey:2019mbc}. We find less optimistic results than those of ref.\,\cite{Hooper:2020evu}.
According to our results, the contribution to $\Delta N_{eff}$ by hot massless gravitons, would escape detection by projected experiments, unless in the case of extremal BH spin values, for all the range of BH masses (that is $10^{-5}-10^9$ g).

In this work we consider BH evaporation as the only production mechanism. The consequences of allowing for other production mechanisms have been recently explored in refs.\,\cite{Gondolo:2020uqv} and \cite{Bernal:2020ili, Bernal:2020bjf}.
For an updated analysis of the possibility that the matter-antimatter asymmetry is due to particles produced by primordial BHs evaporation,
we refer the interested reader to ref.\,\cite{Hooper:2020otu} for GUT baryogenesis, and to ref.\,\cite{Perez-Gonzalez:2020vnz} for leptogenesis.

The paper is organized as follows. In sec.\,2 we introduce our notation and review the basics for primordial Kerr BHs. In sec.\,3, the evaporation process for Kerr BHs is discussed. In sec.\,4 we show the dynamics of the relevant densities from formation to evaporation, while in sec.\,5
we calculate the distribution of the momentum of the emitted particle at the time of evaporation, for the various particle spins. The number and energy densities at evaporation are calculated in sec.\,6, and the upper limit on the DR mass is reviewed in sec.\,7. The contributions to DM and DR are studied in secs.\,8 and 9 respectively.
We draw our summary in the conclusive sec.\,10.

In order to have a better control of our formulas for dimensional analysis and numerical computations, we do not use natural units.

\section{Preliminaries on primordial BHs}

The formation from early Universe inhomogeneities of collapsed objects\,\cite{ZelNov:1967}, later called primordial BHs, was considered in refs.\,\cite{Hawking:1971ei, Carr:1974nx, Hawking:1974sw}.
Since inflation removes all pre-existing inhomogeneities, any cosmologically interesting primordial BH density has however to be created after inflation.
We refer to\,\cite{Carr:2020gox, Khlopov:2013ava} for reviews of the various mechanism proposed, with proper references to the associated literature.
In the following, we present our notations and review the relevant formulas for primordial Kerr BHs formation and dynamics.

\subsection{Radiation dominated era}

According to the first Friedmann equation (neglecting the curvature and cosmological constant terms),
the early Universe evolution is described by
\beq
 \left( \frac{\dot a}{a} \right)^2   \equiv H(t)^2 =\frac{8 \pi G}{3 }   \rho(t)\,,
\label{eq-F1}
\eeq
where $a(t)$ is the scale factor, $H(t)$ is the Hubble parameter, $\rho(t)$ is the mass density of the Universe
and $G$ is the Newton gravitational constant, $G \simeq 6.674 \times 10^{-11} \,\rm{ m^3 / (kg \,s^2)}$.
The Planck mass is $M_{Pl}= \sqrt{ \hbar c / G } \approx 1.221 \times 10^{19}\, \rm{GeV}/c^2  \approx  2.176 \times 10^{-8} $\,kg.

In the early hot and dense Universe, it is appropriate to assume an equation of state corresponding to a fluid of radiation (or relativistic particles).
During radiation domination, $\rho \propto a^{-4}$, $a(t) \propto t^{1/2}$, and
\beq
H(t)= \frac{1}{2 t }\,\,.
\label{eq-raddom}
\eeq
At relatively late times, non-relativistic matter eventually dominates the mass density over radiation.
A pressureless fluid
leads to the expected dependence $\rho \propto a^{-3}$, $a(t) \propto t^{2/3}$, and
\beq
H(t) = \frac{2}{3 t} \,.\,
\label{eq-matdom}
\eeq

The radiation mass density (at high temperatures) can be approximated by including only those particles which are in thermal equilibrium and have masses below the temperature $T_R$ of the radiation bath:
\beq
 \rho_R(t) = \frac{\pi^2 g_*(t)  } {30}  \frac{(k_B T _R(t))^4 }{(\hbar \,c)^3 \,c^2} \,\,\, , \,\,\, \,\,
 g_*(t)= \sum_B g_B + \frac{7}{8}\sum_F g_F \,\,\, ,
 \label{eq-rhoRT}
\eeq
where $k_B$ is the Boltzmann constant, $k_B \simeq 8.617 \times 10^{-5} $ eV/K,
$\hbar$ is the reduced Planck constant, $\hbar=6.582 \times 10^{-16}$ eV s,
$c$ is the velocity of light in vacuum, $c=2.998 \times 10^{8}$ m/s,
and $g_{B(F)}$ is the number of degrees of freedom (dof) of each boson (fermion).

Below the electron mass,
only the photon ($g_\gamma=2$) and three light left-handed neutrinos contribute,
so that $g_*(t)=7.25$.
Below the muon mass,
also the electron (and the positron) has to be included,
so that $g_*(t)=10.75$.
For the full SM, here defined including three light left-handed neutrinos, $g_*(t)= 106.75$.
At higher temperatures, $g_*(t)$ will be model-dependent.
Including the (massless) graviton, ($g_G=2$), has the effect of adding $2$ units to the previously mentioned values of $g_*(t)$.

\subsection{Formation of primordial BHs}

As reviewed for instance in ref.\,\cite{Carr:2020gox}, if a primordial BH forms at the time $t_f$ during the radiation dominated era,
typically its mass is close to the value enclosed by the particle horizon near the end of inflation:
\beq
M_{BH} =  \gamma   \frac{4 \pi}{3}  \rho_R(t_f) \left( 2\, c \,t_f \right)^3 =\gamma   \frac{4 \pi}{3}  \rho_R(t_f)  \left( \frac{ c }{H(t_f)}   \right)^3\,,
\label{eq-MPBH}
\eeq
where $\gamma \lesssim 1$ is a numerical factor that depends on the details of the gravitational collapse,
$\rho_R(t_f)$ and $H(t_f)$ are respectively the radiation density and the Hubble parameter at the formation of the BH,
and in the last equality we used eq.\,(\ref{eq-raddom}).
Using eq.\,(\ref{eq-F1}), we can also write
\beq
 M_{BH} =   \frac{\gamma}{2} \frac{(M_{Pl}c^2) ^2}{ \hbar \,H(t_f)}\frac{1}{c^2} \approx  \gamma   \frac{10^{10} \,\rm{GeV}} {\hbar \,H(t_f)}  10^4 \,\rm{g}  \gtrsim \frac{\gamma}{3} \,{\rm  g} \, ,
\label{eq-PBH}
\eeq
where the last lower bound follows from the fact that CMB observations put a naive upper bound (which does not apply to all inflationary models) on the Hubble scale during inflation, $\hbar H_I \lesssim 3 \times 10^{14}$ GeV at $95\%$ C.L. \cite{Akrami:2018odb}, and $H(t_f) \lesssim H_I$.
In the literature the value $\gamma=1/(3\sqrt{3})\approx 0.2$ is usually taken as reference value\,\cite{Carr:2020gox}; in this case the lower limit would become $M_{BH} \gtrsim 0.07$\,g.
In any case, the BH mass should be larger than the Planck mass, $M_{BH}\gtrsim 10^{-5}$\,g.
As is well known, there is also an upper bound, $M_{BH} \lesssim 10^9$\,g,
because the primordial BH evaporation products have negative effects on nucleosynthesis, see e.g. ref.\,\cite{Keith:2020jww}.
The range of primordial BH masses between these bounds is at present generically unconstrained\,\cite{Carr:2020gox}.

Recalling eq.\,(\ref{eq-raddom}), the primordial BHs formation time is easily found from eq.\,(\ref{eq-PBH}) to be
\beq
\frac{t_{f}}{\hbar}=\frac{1} {\gamma}  \frac{M_{BH} c^2}{ (M_{Pl} c^2)^2  } \,.
\label{eq-tfBH}
\eeq
As for the radiation temperature at formation,
combining eqs.\,(\ref{eq-rhoRT}), (\ref{eq-F1}) and (\ref{eq-PBH}), we have
\beq
k_B T_R(t_{f})
 = \left(  \frac{45 \gamma^2  } {16 \pi^3 g_*(t_{f}) }   \right)^{1/4}  \left( \frac{M_{Pl}  }{M_{BH}} \right)^{1/2}\, M_{Pl} c^2 \,.
 \label{eq-Tf}
\eeq
The temperature and the time at formation of primordial BHs, as a function of the their mass at formation, are plotted {\it e.g.} in fig.\,1 of ref.\,\cite{Masina:2020xhk}.

It is useful to introduce the parameter $\beta$ defined as the BH over the radiation mass density at the formation time
\beq
\beta= \frac{\rho_{BH}(t_f)}{\rho_R(t_f)}= M_{BH} \frac{n_{BH}(t_f)}{\rho_R(t_f)} \, ,
\label{eq-defbeta}
\eeq
where $n_{BH}(t_f)$ is the primordial BH number density at formation, whose
explicit expression can be obtained by combining eqs.\,(\ref{eq-defbeta}) and (\ref{eq-MPBH})
with eq.\,(\ref{eq-F1}), so that
\beq
n_{BH} (t_f)
= \beta \, \gamma^2 \, \frac{3}{32 \pi} \frac{(M_{Pl} c^2)^6}{(M_{BH} c^2)^3 (\hbar c)^3} \,\,.
\label{eq-nBHtf}
\eeq

\subsection{Kerr primordial BHs }

Let $J$ be the angular momentum
of the rotating (uncharged) Kerr BH.
While the mass of the BH can take any positive value,
there is an upper limit on $J$,
\beq
\frac{J}{\hbar} <  \frac{ M_{BH}^2}{M_{Pl}^2
\,\, .
\eeq
This suggest to define a dimensionless spin parameter $a_*$,
\beq
a_*= \frac{J}{\hbar }\frac{ M_{Pl}^2}{M_{BH}^2}
\,\, ,
\eeq
such that $0<a_*<1$. Extremal Kerr BHs are characterized by having $a_*$ very close to $1$, let say $a_*>0.9$ for definiteness.

The size of a Schwarzschild ($a_*=0$) BH, as determined by the radius of the event horizon, is proportional to its mass through
\beq
r_S =2 \hbar c \, \frac{ M_{BH} c^2}{ (M_{Pl} c^2)^2 } \,\,
\eeq
For a BH with nonzero angular momentum, the radius of the event horizon, $r_+$, is smaller than the Schwarzschild radius
\beq
\frac{r_+}{r_S} =  \frac{ 1+ \sqrt{ 1-a_*^2} }{2}    \,\,\, ,
\eeq
until an extremal BH could have an event horizon close to $r_S/2$.

\section{Evaporation of Kerr primordial BHs }

Here we review the basic formulas describing the evaporation of a Kerr BH \cite{Kerr:1963ud, Page:1976ki},
following the notations of refs.\,\cite{Arbey:2019jmj, Arbey:2019mbc}.

Consider a Kerr BH of mass $M_{BH}(t)$ (we neglect the time dependence only when we refer to the formation time).
The Hawking radiation mimics thermal emission from a blackbody with a temperature $T_{BH}(t)$,
given by\,\cite{Hawking:1974sw, Page:1976ki,Page:1976df}
\beq
k_B \,T_{BH}(t)
= \frac{ 1}{8 \pi }   \frac{(M_{Pl} c^2)^2 }{M_{BH}(t) c^2}  \, \frac{2}{ 1 + \frac{1}{ \sqrt{1- a_*(t)^2}} }
 \,\,.
\label{eq-TBH}
\eeq
In the following, we denote by $T_{BH}$ the Hawking temperature at formation, $T_{BH}=T_{BH}(t_f)$.
For the Schwarzschild case, for which $a_*=0$, the formation temperature is denoted by $T^S_{BH}$.

As discussed e.g. in \cite{Page:1976ki}, at the time $t$, such a hole emits particles of type $i$, spin $s_i$ and total energy between $(E, E+dE)$ at a rate, per degree of freedom (dof), given by
\beq
\frac{1}{g_i} \frac{d^2N_i}{dt \,dE} =   \frac{d^2N}{dt \,dE}
=  \frac{1}{2 \pi \hbar } \sum_{\ell,m} \Gamma_{s_i \ell m}(E,M_{BH}(t),a_*(t))
\frac{1}{e^\frac{E'}{k_B T_{BH}(t)} -(- 1)^{2 s_i}} \,\, ,
 \label{eq-d2NdtdE}
 \eeq
where $E'=E-m \Omega$ is the total energy of the particles,
taking into account the BH horizon rotation velocity on top of the total energy $E^2= p^2 c^2+ m_i^2 c^4$ (where $m_i$ is the particle mass),
$m$ is the particle angular momentum projection $m \in [-\ell, +\ell]$, and
$\Omega= \hbar c \,a_*   /(2 r_+  ) $.
The greybody factors $\Gamma_{s_i \ell m}$ are dimensionless absorption probabilities for the emitted species
(in general functions of $E$, $M_{BH}(t)$, $a_*(t)$ and the particle's internal dof and rest mass),
and $g_i$ are the internal dof of the $i$-th particle, which account for the polarization and color dof.

Let us consider in some detail the SM. For the Higgs boson ($s=0$), $g_{h^0}=1$. For the massless ($s=1$) photon and the 8 gluons, $g_\gamma =2$ and  $g_g =16$.
For the massive ($s=1$) W$^\pm$ and Z, $g_{W^+} = g_{W^-} =g_{Z} = 3$. As for the fermions ($s=1/2$): the charged leptons, being Dirac fermions, have
$g_e=g_\mu=g_\tau =4$; the neutrinos have $g_{\nu_e}=g_{\nu_\mu}=g_{\nu_\tau}=2(4)$ in the case they are Majorana (Dirac) particles, respectively;
the quarks have $g_u=g_c=g_t=g_d=g_s=g_b=12$. Finally, one might also include the massless graviton ($s=2$), with $g_G=2$.

\subsection{Rate of mass and angular momentum loss }

The rate of mass loss for an evaporating BH is proportional to the total power emitted
\beq
\frac{c^4 M_{Pl}^4}{\hbar} \frac{f(M_{BH},a_*)}{M_{BH}^2}
=-c^2 \frac{dM_{BH}}{dt} =  \frac{dE}{dt}
= \sum_i  \int_0^\infty  dE \, E\,\frac{d^2N_i}{dt\, dE} \,\,
\,\,.
\label{eq-Mloss}
\eeq
where, to parametrize this, Page \cite{Page:1976df,Page:1976ki} introduced the adimensional Page function, $f(M_{BH},a_*)$.

For the evolution of the BH angular momentum, Page introduced the adimensional function $g(M_{BH},a_*)$ such that
\beq
M_{Pl}^2 c^2\, g(M_{BH},a_*) = -\frac{M_{BH}}{a_*} \frac{dJ}{dt}  = \frac{M_{BH}}{a_*} \sum_i \int dE\, j_i\, \frac{d^2N_i}{dt dE}  \,\, ,
\eeq
where $j_i$ is the angular momentum taken by the $i$-th particle, $j_i = m \hbar$.
The time dependence of $M_{BH}$ and $a_*$ are left understood in the above equations.

Inverting these equations and using the definition of $a_*$, one obtains the differential equations governing the mass
and spin of a Kerr BH
\beq
\frac{dM_{BH}}{dt} =-    \frac{c^2 M_{Pl}^4}{\hbar}  \frac{f(M_{BH},a_*)}{M_{BH}^2} \,\, ,
\label{eq-fK}
\eeq
\beq
\frac{da_*}{dt}
= a_*  \frac{ (M_{Pl}  c^2)^4}{\hbar (M_{BH} c^2)^3}  \left(2 f(M_{BH},a_*) - g(M_{BH},a_*) \right) \,\, .
\label{eq-gK}
\eeq

\begin{figure}[h!]
\vskip .0 cm
 \begin{center}
 \includegraphics[width=7.7cm]{fig-fg}  \,\, \includegraphics[width=7.6cm]{fig-MKerr}
 \end{center}
\caption{\baselineskip=15 pt \small
Left: The Page functions $f$ and $g$ as a function of $a_*$, within the SM (including gravitons) and for BHs with initial mass $M_{BH}\lesssim 10^{10}$ g.
Right: Time dependence of the mass (solid) and spin (dashed) of a Kerr BH in the SM (including gravitons), for $a_*=0, 0.5, 0.8, 0.9,0.97$ and for any value of $M_{BH}\lesssim 10^{10}$ g.  }
\label{fig-K}
\vskip .2 cm
\end{figure}

For the SM (including gravitons), the functions $f(M_{BH},a_*)$ and $g(M_{BH},a_*)$ are constant over the whole range of BH masses we are interested in, namely $(10^{-5}-10^9)$ g, while their dependence on the spin parameter $a_*$ is shown in the left panel of fig.\,\ref{fig-K}.
We calculated the $f(M_{BH},a_*)$ and $g(M_{BH},a_*)$ Page functions using the code BlackHawk\,\cite{Arbey:2019mbc}, and
we checked that they are consistent with previous numerical estimates\,\cite{Page:1976ki,Dong:2015yjs,Taylor:1998dk}.
In particular, in the Schwarzschild case, one has $f_S(M_{BH})=f(M_{BH},0) = 4.27 \times 10^{-3}$.

\subsection{BH lifetime}

Given any initial values for the BH mass and spin, the functions $f(M_{BH},a_*)$ and $g(M_{BH},a_*)$ allow one to obtain the time evolution of the BH mass and spin, by using eqs. (\ref{eq-fK}) and (\ref{eq-gK}).

For a Schwarzschild BH, the lifetime equals the time of the BH evanescence, $t_{ev}^S$, so that
\beq
\frac{t_{ev}^S}{\hbar} = \frac{1}{c^2 M_{Pl}^4 }  \int_0^{M_{BH}} dM  \frac{M^2}{f(M,0)}
= \frac{1}{3  f_S(M_{BH})} \frac{(M_{BH} c^2)^3}{(M_{Pl} c^2)^4} \,\, ,
\label{eq-tevBH}
\eeq
where in the last equality we assumed $f(M,0)$ to be constant over the BH lifetime (as in the SM).
For Schwarzschild, one thus obtains a simple time dependence for the ratio of the BH mass at time $t$ over its initial value,
\beq
\frac{M^S_{BH}(t)}{M_{BH} } = \left( 1- \frac{t}{t_{ev}^S}\right)^{1/3} \, ,
\label{eq-MBHt}
\eeq
as illustrated in the right panel of fig.\,\ref{fig-K}.

For the more general Kerr case, considering for definiteness the SM (including gravitons), we display in fig.\,\ref{fig-K} the time dependence of the mass ratio $M_{BH}(t)/M_{BH}$ (solid lines) and of the spin $a_*(t)$ (dashed lines). In particular, we select some representative initial spin values, $a_*=0.5,0.8,0.9,0.97$.
The results hold for any value of $M_{BH}\lesssim 10^{10}$ g.
The lifetime is more and more reduced the higher is the initial spin of the BH:
this means that the evaporation time in the Kerr case is smaller than in the Schwarzschild case. To account for the shortening in the lifetime, it is useful to introduce the parameter $\alpha_K$ such that
\beq
t_{ev} = \alpha_K  t_{ev}^S \,\, .
\label{eq-aK}
\eeq

From fig.\,\ref{fig-K}, one might suppose that a Kerr BH behaves like a Schwarzschild BH in the final stages of its lifetime.
In particular, in his last period, the time dependence of a Kerr BH mass is similar to the one of a Schwarzschild BH
with an initial effective mass given by $M^{eff}_{BH}=\alpha_K^{1/3} M_{BH}$.

It is not difficult to study the effect of adding beyond SM particles, like for instance axions and right-handed neutrinos.
As far as the number of dof of the additional particles is small with respect to the SM one, the inclusion of such additional particles does not change appreciably the results of fig.\,\ref{fig-K}.

\section{From formation to evaporation}

\subsection{Radiation vs BH domination}

Let us define the ratio $f(t)=\rho_{BH}(t)/\rho_R(t)$.
Since $\rho_{BH}(t) \propto 1/a(t)^3$, while $\rho_R(t)\propto 1/a(t)^4$,
such ratio increases as the scale factor, $f(t)\propto a(t)$.
It is thus possible that BHs come to dominate the energy content of the universe
before they completely evaporate \cite{Barrow:1990he, Baumann:2007yr, Fujita:2014hha}:
this situation is referred to as BH domination.
The scenario in which evaporation takes place before BH domination might occur is referred to as radiation domination.

We define $\bar \beta$ to be the maximum value of $\beta$
corresponding to radiation domination, namely the value of $\beta$ leading to ${f(t_{ev})} = 1$;
this value can be obtained from the following relation,
\beq
 {\bar \beta} = \frac{f(t_f)}{f(t_{ev})} = \frac{a(t_f)}{a(t_{ev})}
 = \left( \frac{t_f}{\alpha_K t^S_{ev}} \right)^{1/2}
=  \frac{1}{\alpha_K^{1/2}}   \left(     \frac{ 3 f_S(M_{BH})}{\gamma}  \right)^{1/2}  \frac{M_{Pl} } {M_{BH} }
\,\, ,
\label{eq-barbeta}
\eeq
where we used eqs.\,(\ref{eq-aK}), (\ref{eq-tfBH}) and (\ref{eq-tevBH}).
For all the values of $\beta \lesssim \bar \beta$,
the primordial BHs evaporate before they come to dominate the matter content of the Universe,
and the increase in the scale factor is simply given by
\beq
\frac{a(t_f)}{a(t_{ev})} = {\bar \beta}  \,\, .
 \label{eq-summR}
\eeq

For all the values of $\beta \gtrsim \bar \beta$, BH domination occurs.
We have thus to consider: first, the radiation dominated period from the formation time, $t_f$, to the time when BHs start to dominate, $t_{BH}$, such that $f(t_{BH})= 1$; and second, the matter dominated period from $t_{BH}$ to $t_{ev}$.
The first period is characterized by the following increase in the scale factor
\beq
{\beta} = \frac{f(t_f)}{f(t_{BH})} = \frac{a(t_f)}{a(t_{BH})} = \left( \frac{t_f}{t_{BH}} \right)^{1/2} \,\, ,
\eeq
while the second period is characterized by
\beq
\frac{a(t_{BH})}{a(t_{ev})} = \left( \frac{t_{BH}}{t_{ev}} \right)^{2/3} =  \left( \frac{1}{\beta^2} \frac{t_{f}}{t_{ev}} \right)^{2/3}
= \left( \frac{1}{\beta^2} \frac{t_{f}}{ \alpha_K t^S_{ev}} \right)^{2/3}

 =\left( \frac{\bar \beta}{\beta}  \right)^{4/3}\,\, ,
\eeq
so that, putting together,  the total increase in the scale factor is given by
\beq
\frac{a(t_f)}{a(t_{ev})} = \frac{a(t_f)}{a(t_{BH})} \frac{a(t_{BH})}{a(t_{ev})}
= \frac{\bar \beta^{4/3}}{\beta^{1/3}}= \bar \beta  \left( \frac{\bar \beta}{\beta}  \right)^{1/3} \,\, .
 \label{eq-summBH}
\eeq

\subsection{BH density at evaporation}

The BH number density at the time of evaporation is related to the one at formation by
\beq
n_{BH} (t_{ev}) = n_{BH} (t_f) \left( \frac{a(t_f)}{a(t_{ev})}\right)^3 \,.
\label{eq-nBHtev}
\eeq
Using the equation above together with eqs.\,(\ref{eq-nBHtf}), (\ref{eq-barbeta}) and (\ref{eq-summR}), one obtains for radiation domination
\beq
n_{BH}(t_{ev}) = \frac{1}{\alpha_K^{3/2}} \,   \beta \,\gamma^{1/2} \,\,   \, \frac{3}{ 32 \pi} \,
  { (3 f_S(M_{BH}))^{3/2}} \,  \left( \frac{M_{Pl} } {M_{BH} }\right)^6 \,\frac{(M_{Pl} c^2)^3}{(\hbar c)^3} \, \,\, ,
 \label{eq-nBHtevR}
\eeq
while, using eq.\,(\ref{eq-summBH}), one obtains for BH domination
\beq
n_{BH}(t_{ev}) =   \frac{1}{\alpha_K^{2}}  \, \, \frac{3}{32 \pi} \,
  ( 3 f_S(M_{BH}) )^2  \,  \left( \frac{M_{Pl} } {M_{BH} }\right)^7 \,\frac{(M_{Pl} c^2)^3}{(\hbar c)^3} \, \,\, ,
     \label{eq-nBHtevBH}
\eeq
which displays an increase with respect to radiation domination by the factor $\bar \beta/\beta$.

\subsection{Radiation temperature at evaporation}

One can grossly assume that all the energy density stored in the BHs goes, after their evaporation, into the radiation energy density of the SM particles and of possibly beyond SM particles emitted by the BH:
\beq
\rho_{BH}(t_{f}) \left( \frac{a(t_f)}{a(t_{ev})} \right)^3 =\rho_{BH}(t_{ev}) \approx \rho_{SM}(t_{ev}) + \rho_{BSM}(t_{ev}) \,\, .
\eeq
Both the SM and beyond SM particles are emitted with a spectrum of momenta that is not thermal.
However, the SM particles produced in the evaporation of the BH rapidly thermalize as soon as they are emitted.
In this work we assume that the beyond SM particles are stable and interact feebly or only gravitationally, so that they never come in thermal equilibrium.

In the case of radiation domination,
the radiation energy density from the SM particles emitted by the BH is negligible with respect to the radiation present since the formation of the BH,
 $ \rho_{BH}(t_{ev})  << \rho_{R}(t_{ev})$.
Combining eqs. (\ref{eq-F1}), (\ref{eq-raddom}) and eq.\,(\ref{eq-aK}),
\beq
\frac{8 \pi G}{3} \rho_R(t_{ev})  = \frac{1}{4 t_{ev}^2}= \frac{1}{4 \alpha_K^2 {t^S_{ev}}^2} \,\, ,
\eeq
and using eqs.\,(\ref{eq-rhoRT}) and (\ref{eq-tevBH}), one obtains
\beq
k_B T_R(t_{ev}) = \frac{1}{\alpha_K^{1/2}} \,
 \left(   3 f_S(M_{BH}) \right)^{1/2}
 \left( \frac{ 45  }{16 \pi^3 g_{*}(t_{ev})}     \right)^{1/4}  \left( \frac{     M_{Pl} }{  M_{BH}  }  \right)^{3/2} (M_{Pl} c^2)
 \approx  \frac{1}{\alpha_K^{1/2}} \, k_B T^S_R(t_{ev})\,\, ,
\label{eq-TevR}
\eeq
namely that the temperature in the Kerr case is slightly higher than in the Schwarzschild case:
notice indeed that, since $\alpha_K$ is in general bigger than $1/2$ (see right panel of fig.\,\ref{fig-K}), the function $g_{*}(t_{ev})$ is very similar to what it is in the Schwarzschild case, here denoted by $g_{*}(t^S_{ev})$. The values of $g_{*}(t^S_{ev})$, as a function of the BH mass, are shown in the right panel of fig. 3 of ref.\,\cite{Auffinger:2020afu}.

In the case of BH domination, by definition, $\rho_{BH}(t_{ev})  >> \rho_{R}(t_{ev})$.
If the fraction of beyond SM particles is not enormous, one also has $\rho_{SM}(t_{ev}) \approx \rho_{BH}(t_{ev})$.
The beyond SM particles remain non thermal if they interact feebly or only gravitationally.
At the contrary, the SM particles produced in the evaporation of the BH rapidly thermalize as soon as they are emitted, so that
the radiation energy density (and thus the radiation temperature) gets a sudden increase, going up to $\rho_R(t_{ev}) \approx \rho_{BH}(t_{ev})$ after thermalization.
Assuming the period of matter domination by BHs is respectively short or long with respect to the first period of radiation domination, and combining eqs. (\ref{eq-F1}) and (\ref{eq-matdom}), we have
\beq
\frac{8 \pi G}{3} \rho_{R}(t_{ev})
=\left( 1\, {\rm or}\, \frac{16}{9} \right) \frac{1}{4 t_{ev}^2} \,\, .
\eeq
For a short or long period of matter domination by BHs, the radiation temperature after evaporation gets slightly enhanced with respect to radiation domination, by a factor going from $1$ up to $({16}/{9})^{1/4}\approx 1.15$.

\subsection{BH number to entropy density at evaporation}

For later convenience it is useful to introduce $Y_{BH}(t)$, the number to entropy density of BHs at time $t$
\beq
Y_{BH}(t) =  \frac{n_{BH}(t)}{s(t)} \, ,
\label{eq-YBH}
\eeq
where the entropy density is defined as
\beq
s(t) = \frac{ 2 \pi^2 g_{*,S}(t) }{45} \frac{(k_B T_R(t))^3}{(\hbar c )^3} \,.
\label{eq-s}
\eeq
The difference between $g_{*}(t)$ and $g_{*,S}(t)$ can in general be neglected.

For radiation domination, using eqs.\,(\ref{eq-nBHtevR}) and (\ref{eq-TevR}),
we have that the number-to-entropy density in the Kerr and Schwarzschild cases are equal
\beq
Y_{BH}(t_{ev}) =\beta \,\gamma^{1/2} \,\,   \, \frac{3}{ 4 } \,
 \left( \frac{ 45  }{16 \pi^3 g_{*}(t_{ev})}     \right)^{1/4}   \left( \frac{M_{Pl} } {M_{BH} }\right)^{3/2}

 \,\, ,
 \label{eq-YBHR}
\eeq
as the dependence on $\alpha_K$ disappears.
For a short period of matter domination by BHs, using eqs.\,(\ref{eq-nBHtevBH}) and (\ref{eq-TevR})
\beq
Y_{BH}(t_{ev}) = \,  \,
\frac{1}{\alpha_K^{1/2}}    ( 3 f_S(M_{BH}) )^{1/2}  \,   \,
\, \frac{3}{4 }\, \left( \frac{ 45  }{16 \pi^3 g_{*}(t_{ev})}     \right)^{1/4} \,\left( \frac{M_{Pl} } {M_{BH} }\right)^{5/2} \,\, ,
 \label{eq-YBHBH}
\eeq
which displays an increase with respect to radiation domination by the factor $\bar \beta/\beta$.

\section{Momentum distribution at evaporation}

Eq. (\ref{eq-d2NdtdE}) gives the instantaneous spectrum of the particles of type $i$ emitted by a single BH.
The maximum of the energy distribution is at about $E \sim k_B T_{BH}$.
If the particle is sufficiently light, as we are going to assume in the following, the ultra relativistic limit applies, $E \approx c p > m_i c^2$.

The distribution of the momentum at evaporation for the particle of type $i$, normalized per dof, is a superposition of all the instantaneous distributions, each redshifted appropriately from its time of emission $t_{em}$ (see \emph{e.g.} ref.\,\cite{Bugaev:2000bz})
\beq
\frac{1}{g_i} \frac{dN_i}{d(cp)}(t_{ev})
=\int_{t_{em}}^{t_{ev}} dt \,  \frac{d^2 N}{ dt\,d(c p(t)) }  \left( \underbrace{c p(t_{ev}) \frac{a(t_{ev})}{a(t)} }_{cp(t)}, T_{BH}(t) ,a_*(t)\right) \,\,  \frac{a(t_{ev})} {a(t)} \, .
\label{eq-Fev}
\eeq
Notice that $t_{em}$ might be larger than $t_f$ if the initial BH temperature is smaller than the particle mass but,
since we are interested in light DM, $t_{em}= t_f$.

For radiation domination, the ratio of scale factors to be put in eq.\,(\ref{eq-Fev}) is
\beq
 \frac{a(t_{ev})}{a(t)} = \left( \frac{t_{ev}}{t} \right)^{1/2} \,.
 \eeq

For BH domination, the integral of eq.\,(\ref{eq-Fev}) should be split into two contributions, corresponding to a first period of radiation domination, and a second of BH domination.
For the second period of BH domination, starting at $t_{BH}=t_f/\beta^2$ and ending at $t_{ev}$,
the ratio of scale factors to be put in the integrand is
\beq
 \frac{a(t_{ev})}{a(t)} = \left( \frac{t_{ev}}{t} \right)^{2/3} \, \,,
\eeq
while for the first period of radiation domination, starting at $t_f$ and ending at $t_{BH}$,
the ratio of scale factors to be put in the integrand is rather
\beq
 \frac{a(t_{ev})} {a(t)} =  \frac{a(t_{ev})} {a(t_{BH})} \frac{a(t_{BH})} {a(t)}
 = \left( \frac{t_{ev}}{t_{BH}} \right)^{2/3} \left( \frac{t_{BH}}{t} \right)^{1/2}  \, .
\eeq
For short (long) BH domination, clearly the dominant contribution comes from the first (second) period.

It is useful to define the adimensional momentum
\beq
x(t_{ev}) \equiv \frac{c p(t_{ev})}{k_B T^S_{BH} } \, ,
\eeq
with the related adimensional momentum distribution at evaporation
\beq
\tilde F_{s_i}(x(t_{ev}))  \equiv \frac{(k_B T^S_{BH})^3}{(M_{Pl} c^2)^2} \frac{1}{g_i} \frac{dN_i}{d(cp)}(t_{ev})
\,\, ,
\label{eq-tFev}
\eeq
which has the nice property of depending only on the particle spin $s_i$ and the BH spin $a_*$, while being independent of the BH mass.

\begin{figure}[h!]
\vskip .0 cm
 \begin{center}
\includegraphics[width= 7.6 cm]{fig-tFev-BHS.pdf}   \,\,
  \includegraphics[width= 7.6 cm]{fig-tFev-BHK5.pdf}   \,\,
 \vskip .5 cm  \includegraphics[width= 7.6 cm]{fig-tFev-BHK8.pdf}
 \,\,  \includegraphics[width= 7.6 cm]{fig-tFev-BHK9.pdf}
\end{center}
\caption{\baselineskip=15 pt \small
Adimensional function $\tilde F(x(t_{ev)})$, as defined by eq.\,(\ref{eq-tFev}), assuming radiation domination with $\beta=\bar \beta$ (solid) or full BH domination (dashed), calculated using \texttt{BlackHawk}\,\cite{Arbey:2019mbc} within the SM and for various particle spins, as indicated.
Top left: Schwarzschild case. Top right and bottom: Kerr case with $a_*=0.5$, $a_*=0.8$ and $a_*=0.9$.}
\label{fig-distr-ev}
\vskip .2 cm
\end{figure}

For the Schwarzschild case, previously studied in ref.\,\cite{Auffinger:2020afu},
the quantity $\tilde F_{s_i}(x(t_{ev}))$ derived from \texttt{BlackHawk}\,\cite{Arbey:2019mbc} is shown in the top left panel of fig.\,\ref{fig-distr-ev},
assuming radiation domination with $\beta = \bar \beta$ (solid) and full (namely a long period of) BH domination (dotted).
The suppression due to the different values of the spin is manifest.
The other panels show the same for the Kerr case, taking different initial spins, $a_*=0.5, 0.8, 0.9$, as indicated.

Since the particle momentum scales in time as the inverse of the scale factor, the contribution at small $x(t_{ev})$ is dominated by evaporation in the first period of the BH lifetime, while the contribution at large $x(t_{ev})$ is dominated by evaporation in the last period of the BH lifetime,
for which the momentum is not much suppressed.
The contribution at small $x(t_{ev})$ is thus particularly sensitive to the Kerr regime: the higher is the value of BH spin parameter $a_*$,
the more the contribution from particles with high spin, like $s=2$ and $s=1$, gets enhanced. This fact is well known, in particular for gravitons\,\cite{Page:1976ki}. For large $x(t_{ev})$ the four plots of fig.\,\ref{fig-distr-ev} are instead similar, because the Kerr BH has already slowed down its rotation,
so that the evaporation at late times resembles the Schwarzschild case.

\section{Particle densities at evaporation}
\label{sec-PD}

In this section we calculate the particle number and energy densities for a Kerr BH, which will be useful in the following sections.

\subsection{Number densities at evaporation}

The density at the evaporation time of the particles of type $i$ emitted in the evaporation of the BHs, is given by
\beq
n_i (t_{ev}) =\int dE \,  \, \frac{dn_i}{dE} (t_{ev}) ={n_{BH}(t_{ev})} \, \int dE\, \, \frac{dN_i}{dE}(t_{ev}) =
 n_{BH}(t_{ev}) \,N_i \,\, ,
\eeq
where
$N_i$ is the total number of $i$ particles produced in the evaporation of a single BH.

For the relativistic regime, the calculation of $\frac{dN_i}{d(cp)}(t_{ev})$ was done in the previous section, eq.\,(\ref{eq-Fev}).
Using also eq.\,(\ref{eq-tFev}), we have
\beq
N_i
=(8 \pi)^2 \frac{M_{BH}^2}{M_{Pl}^2} \,  g_i \,  {\tilde \phi_{s_i}} \,\,\, , \,\,\,\,\,\,\,\, {\tilde \phi_{s_i}}=\int_0^\infty dx(t_{ev})\,  \tilde F_{s_i}(x(t_{ev})) \,\, .
\label{eq-NXphi}
\eeq
We show in the left panel of fig.\,\ref{fig-tphi} the quantity $\tilde \phi_{s_i}$: notice that it does not depend on the BH mass,
but only on the particle spin $s_i$ and the initial spin of the BH, $a_*$.
It turns out that $\tilde \phi_{s_i}$ is not appreciably different for radiation or BH domination (the difference being at the percent level).
 We can see the remarkable increase in $\tilde \phi_{s_i}$ going from Schwarzschild to extremal Kerr BHs  for $s_i=2$;
 for lower particle spins, the dependence on $a_*$ is instead mild.

It is interesting to compare the results of the left panel of fig.\,\ref{fig-tphi} for $a_*=0$, with an approximation, obtained in the Schwarzschild case by adopting the geometrical optics approximation (that differentiates only bosons and fermions).
Neglecting redshift effects and using the geometrical optics approximation, we have
\beq
 N_i =\int_{t_f}^{t_{ev}} dt\,  \int_0^\infty  dE\,   \frac{d^2N_i}{dt dE} \,\,
 = \frac{1}{2 f_S(M_{BH})}  \frac{27 \zeta(3)}{8^3 \pi^4 }  \,\{1,\frac{3}{4}\} \,g_{i}    \frac{M_{BH}^2 }{M_{Pl}^2 }

 \,\, ,
\label{eq-GOsf}
\eeq
where the first (second) factor in the curly brackets stands for bosons (fermions).
In particular, comparing with eq. (\ref{eq-NXphi}), we identify
\beq
\,  \,  {\tilde \phi_{s_i}} \leftrightarrow \frac{1}{(8 \pi)^2}   \frac{1}{2 f_S(M_{BH})}  \frac{27 \zeta(3)}{8^3 \pi^4 }  \, \,\{1,\frac{3}{4}\}
= \{1.2 , 0.90 \} \,\times \,10^{-4}\,\,,
\eeq
which is quite good in the bosonic case, but not very satisfctory in the fermionic one.

\begin{figure}[h!]
\vskip .2 cm
 \begin{center}
\includegraphics[width= 7.6 cm]{fig-tphi}   \,\,\includegraphics[width= 7.6 cm]{fig-tx}
\end{center}
\caption{\baselineskip=15 pt \small
Dependence on $a_*$ of the quantities $\tilde \phi_{s_i}$ and $\tilde x_{s_i}$, for various particles spins, as indicated.
Radiation domination and BH domination give nearly indistinguishable results.}
\label{fig-tphi}
\vskip .2 cm
\end{figure}

\subsection{Energy densities at evaporation}

The energy density of the particle $i$ at evaporation is
\beq
\rho_i (t_{ev}) =\int dE \,\frac{E}{c^2}  \, \frac{dn_i}{dE} (t_{ev}) =\frac{n_{BH}(t_{ev})}{c^2} \, \int dE\, E\, \frac{dN_i}{dE}(t_{ev})
= \frac{n_{BH}(t_{ev})}{c^2} \,N_i <E_i(t_{ev})> \,\, ,
\label{eq-rhoXev}
\eeq
where $<E_i(t_{ev})>$ is the mean energy of the $i$ particle at evaporation.

In the relativistic regime, we use eqs. (\ref{eq-Fev}) and (\ref{eq-tFev}), to obtain
\beq
N_i <E_i(t_{ev})
= (8 \pi) (M_{BH} c^2)  \, g_i \,  {\tilde x_{s_i}} \,\, , \,\,\,\,\,\,\,  \tilde x_{s_i}= \int_0^\infty dx(t_{ev})\, x(t_{ev})\,  \tilde F_{s_i}(x(t_{ev})) \,\, .
\label{eq-tx}
\eeq
As it was the case for $\tilde \phi_{s_i}$, also $\tilde x_{s_i}$ depends on $a_*$ and the particle spin, but not on the BH mass.

The right panel of fig.\,\ref{fig-tphi} shows the numerical values of $\tilde x_{s_i}$ and its dependence on $a_*$.
The results are obtained for radiation domination, but $\tilde x_{s_i}$ is only marginally different for BH domination.
We can see a remarkable increase for extremal BHs for $s_i=2$ particles, while for lower particle spins the dependence on $a_*$ is mild.

It is interesting to compare the results of the plot for $a_*=0$, with an approximation obtained in the Schwarzschild case and neglecting the redshift effect,
so that, instead of eq.\,(\ref{eq-Fev}), one uses
\beq
 \frac{dN_i}{d(cp)}(t_{ev})
=\int_{t_{em}}^{t_{ev}} dt \,  \frac{d^2 N_i}{ dt\,d(c p(t)) }  \left( {cp(t)}, T_{BH}(t) \right) \,\,   ,
\eeq
and the energy density of eq.\,(\ref{eq-rhoXev}) becomes instead
\beq
\rho_i (t_{ev})
=\frac{n_{BH}(t_{ev})}{c^2} \,  \frac{c^4 M_{Pl}^4}{\hbar}\,\int_{t_{em}}^{t_{ev}} dt \,   \frac{g_i\, f_i(M_{BH}(t))}{M_{BH}(t)^2}
 =n_{BH}(t_{ev})  \, M_{BH}\, \frac{g_i \,f_{i,S}(M_{BH})}{f_S(M_{BH})}  \,\, .
\eeq
where $g_i f_{i,S}(M_{BH})$ is the contribution to the Page function $f_S(M_{BH})$ by the $i$ particle, which in the last equality was
assumed to be constant over the BH lifetime.
Recalling the definition of $\tilde x_{s_i}$, eq. (\ref{eq-tx}),
we can identify
\beq
  \, \tilde x_{s_i}  \leftrightarrow \frac{1}{ 8 \pi} \,\frac{f_{i,S}(M_{BH})}{f_S(M_{BH})} = \{ 7.0,3.9,1.6,0.18\}\times\,  10^{-4}\,\,\,,\,\, {\rm for }\,s=0,1/2,1,2\,\, ,
\eeq
where the numerical values in the right hand side have been obtained within the SM (that is for $M_{BH}<10^{10}$\, g).
Notice the quite good agreement with the numerical values of fig.\,\ref{fig-tphi} for $a_*=0$: this demonstrates that the inclusion of the redshift effect is not dramatic.

In the case of BH domination, from eqs.\,(\ref{eq-rhoXev}) and (\ref{eq-tx}), the fraction of the energy density of the $i$ species with respect to the energy density of radiation (that is all the SM particles) at the evaporation is
\beq
\frac{\rho_i(t_{ev})} {\rho_{BH}(t_{ev})} \approx \frac{\rho_i(t_{ev})} {\rho_{R}(t_{ev})}
= \frac{N_{i} <E_{i}(t_{ev})>}{\sum_{j=R} N_j <E_j(t_{ev})>}
=\frac{g_{i}\, \tilde x_{s_{i}}}{ \sum_{j=R}  g_{j} \,\tilde x_{s_j} } \,\, ,
\label{eq-rhoiR}
\eeq
where the sum over radiation includes all the SM. This is shown the left plot of fig.\,\ref{fig-td}, considering a boson ($g_i=1$), a Weyl fermion ($g_i=2)$,
a massive vector ($g_i=3$), a massive ($g_i=5$) and a massless ($g_i=2$), graviton.
For instance, for a massless (massive) graviton and $a_*=0.7$, we have $\rho_G/\rho_R= 0.20\% (0.50\%)$;
for an extremal BH with $a_*=0.97$, the latter values increase to $1.4\% (3.6\%)$ respectively.

In the case of radiation domination, exploiting the fact that ${\rho_{BH}(t_{ev})} /{\rho_{R}(t_{ev})}=  f(t_{ev})  ={\beta}/{\bar \beta}$, one has
\beq
\frac{\rho_i(t_{ev})} {\rho_{R}(t_{ev})}
= \frac{\rho_i(t_{ev})} {\rho_{BH}(t_{ev})}  \frac{\rho_{BH}(t_{ev})} {\rho_{R}(t_{ev})}
=   \frac{\beta}{\bar \beta}  \frac{g_{i}\, \tilde x_{s_{i}}}{ \sum_{j=R}  g_{j} \,\tilde x_{s_j} } \,\, ,
\label{eq-rhoiBH}
\eeq
namely a global suppression by a factor ${\beta}/{\bar \beta}$ with respect to the results in the left panel of fig.\,\ref{fig-td}.

\begin{figure}[h!]
\vskip .5 cm
 \begin{center}
 \includegraphics[width= 7.6 cm]{fig-DR}   \,\,\,\,\,\,  \,\,\includegraphics[width= 7.6 cm]{fig-td}
\end{center}
\caption{\baselineskip=15 pt \small
Left: The fraction of the energy density of the $i$ species with respect to the energy density of radiation (that is all the SM particles) at the evaporation,
${\rho_i(t_{ev})}/ {\rho_{R}(t_{ev})}$, as a function of $a_*$ and for BH domination. We consider a boson ($g_i=1$), a Weyl fermion ($g_i=2)$,
a massive vector ($g_i=3$), a massive ($g_i=5$) and a massless ($g_i=2$), graviton.
Right: The dependence of $\alpha_K^{1/2} \tilde \delta_{s_X}$ on the BH spin $a_*$, for various particle spins, as indicated.}
\label{fig-td}
\vskip .2 cm
\end{figure}

\section{Viable masses for DR}

In this paper we consider the possibility that, in top of the SM, a particle of the species $X$, with mass $m_X$, is produced in the evaporation of the BH.
If such particle is stable, it would contribute to DM and, if sufficiently light, it would give a significant contribution to DR \cite{Lennon:2017tqq,Hooper:2019gtx,Masina:2020xhk}. Here we generalize to the Kerr case the argument reviewd in ref.\,\cite{Masina:2020xhk} for the Schwarzschild case.

In order for the $X$ particles to give a sizable contribution to DR, their average kinetic energy evaluated at the time of matter radiation equality,
$t_{EQ}$, must exceed their mass: $c p_{EQ} \approx \langle E_X(t_{EQ}) \rangle \gtrsim m_X c^2$.
Using eqs.\,(\ref{eq-NXphi}) and (\ref{eq-tx}), the average kinetic energy of the emitted $X$ particles is\footnote{It is
interesting to compare our numerical result with a previous analytical approximation, valid in the Schwarzschild case\,\cite{Fujita:2014hha},
$c p_{ev}  \approx \langle E_X (t_{ev}) \rangle \approx 6 \, k_B T^S_{BH}$.}
\beq
c p_{ev}  \approx <E_X(t_{ev})>

 =\frac{ \, \tilde x_{s_X} } {\tilde \phi_{s_X}}  \, k_B T^S_{BH} \equiv {\tilde \delta_{s_X}} \, k_B T^S_{BH} \,\, .
 \label{eq-meanE}
\eeq

Since the momentum scales as the scale factor,
\beq
\langle E_X (t_{EQ}) \rangle
\approx \langle E_X (t_{ev}) \rangle \, \frac{a(t_{ev})}{a(t_{EQ})}
= \tilde  \delta_{s_X} \, (k_B T^S_{BH})\frac{1}{\alpha'} \frac{k_B T_R(t_{EQ})}{k_B T_R(t_{ev})} \left( \frac{g_{*,S}(t_{EQ})}{g_{*,S}(t_{ev})} \right)^{1/3} \,\, ,
\label{eq-ETEQ}
\eeq
where in the last equality we assumed entropy conservation from evaporation to matter-radiation equality, $\alpha' (s a^3)_{ev}=(s a^3)_{EQ}$.
Using eq.\,(\ref{eq-TevR}), we find the dependence on the BH mass
\beq
 \frac{k_B T^S_{BH}}{k_B T_R(t_{ev})} ={\alpha_K^{1/2}} \, \frac{1}{8 \pi}  \, \left(   \frac{1}{3 f_S(M_{BH})} \right)^{1/2}
 \left( \frac{16 \pi^3 g_{*}(t_{ev})}{ 45  }     \right)^{1/4}   \left( \frac{     M_{BH} }{  M_{Pl}  }  \right)^{1/2}   \,\, ,
\eeq
so that, assuming $g_{*,S}(t_{EQ}) \approx 3.94$, $ g_{*,S}(t_{ev})\approx 108.75$, taking $k_B T_R(t_{EQ}) \approx 0.75$ eV and $\alpha'=1$,
the condition to contribute significantly to DR becomes
\beq
m_X c^2 \lesssim \langle E_X (t_{EQ}) \rangle
\approx
{\alpha_K^{1/2}}\,  { \tilde \delta_{s_X}} \, \left( \frac{M_{BH}}{1 \,{\rm g}} \right)^{1/2}\,0.11 \,{\rm keV}\, .
\label{eq-EmEQ}
\eeq

The right panel of fig.\,\ref{fig-td} shows the dependence of the quantity $\alpha_K^{1/2} \tilde \delta_{s_X}$ on the BH spin $a_*$.
For a boson in the Schwarzschild case $\tilde \delta_0\approx 4.5$, so that $m_X c^2 \lesssim 0.50$ keV for $M_{BH}=1$ g, in agreement with ref.\,\cite{Masina:2020xhk}.
For the Kerr case, the suppression induced by factor $\alpha_K^{1/2}$ is mild (at most $0.7$ for extremal BH). Including also the factor $\delta_{s_X}$, we can see from the right panel of fig.\,\ref{fig-td} no significant difference with respect to the Schwarzschild for $s=0,1/2$, while the decrease is significant for $s=1$ and especially for $s=2$.

\section{Stable particles as dark matter}

In this section we assume that the $X$ particle is going to provide the full contribution to the DM observed today.

In the Schwarzschild case, it is well known that there are two possible solutions for DM, denoted as "light" and "heavy" DM \cite{Fujita:2014hha},
according to the fact
that the particles are produced during all the BH lifetime or just in its final stages (see e.g. \cite{Masina:2020xhk} and references therein).
If $m_X c^2 < k_B T^S_{BH}$, the DM candidate belongs to the "light" category, otherwise to the "heavy" one.
In particular, the light DM case is subject to strong constraints from structure formation \cite{Fujita:2014hha, Lennon:2017tqq, Masina:2020xhk, Baldes:2020nuv, Auffinger:2020afu}, so that the BH domination scenario is ruled out.
One might guess \cite{Auffinger:2020afu} that the Kerr case would manage to escape such constraints.
This what we study in this section, by generalizing the study of the light DM scenario to Kerr BHs.

The cosmological abundance related to the species $X$ at the present time, $t_0$, is proportional to the present number to entropy density
of such species
\beq
\Omega_{X} =\frac{\rho_{X}   }{\rho_{c}}=\frac{m_{X}  }{\rho_{c}} \frac{n_{X}(t_{0}) }{s(t_{0}) }\, s(t_{0})
=\frac{m_{X} \,s(t_{0})  }{\rho_{c}}   Y_X(t_{0})   \, \,,
\label{eq-OX}
\eeq
where, defining $H= 100\, h \,{\rm km\,s^{-1}\, Mpc^{-1}}$,
\beq
\rho_c =\frac{3 H^2}{8 \pi G} =1.88 \times 10^{-26} \,h^2 \,{\rm \frac{kg}{m^3}} \,.
\eeq
The entropy density now is obtained from eq.\,(\ref{eq-s}) by putting the CMB temperature $T_{CMB}=2.7255$ K \cite{Akrami:2018odb}:
$s(t_{0})= 2891 /{\rm cm^3}$.
Observationally, the cosmological abundance of cold DM has to be $\Omega_c\approx 0.25$.

One can treat evaporation \cite{Baumann:2007yr} as if all particles were produced at a single instant, $t\approx t_{ev}$.
The present number-to-entropy density of a stable particle $X$ produced by evaporation is directly related to the BH abundance at evaporation
 \cite{Baumann:2007yr, Fujita:2014hha} by
\beq
Y_X(t_{0}) =\frac{n_{X}(t_{0}) }{s(t_{0}) }= \frac{1}{\alpha}\frac{n_{X}(t_{ev}) }{s(t_{ev}) }
= \frac{1}{\alpha} N_{X} \frac{n_{BH}(t_{ev})}{s(t_{ev})} = \frac{1}{\alpha} N_{X}  Y_{BH}(t_{ev})\,,
\label{eq-YXtnow}
\eeq
where $\alpha$ parametrizes a possible entropy production after evanescence, $\alpha (s a^3)_{ev}=(s a^3)_{0}$,
and $N_{X}$ is the number of $X$ particles produced in the evaporation of a single BH\footnote{It is reasonable to assume that entropy is conserved from matter-radiation equality to the present time, so that\,$\alpha \approx \alpha'$.}.
The detailed calculation of $N_X$ was carried out in sec.\,\ref{sec-PD}, while $Y_{BH}(t_{ev})$ can be read from eqs.\,(\ref{eq-YBHR}) and (\ref{eq-YBHBH}).

As reference DM case, we consider a scalar boson (with $g_X=1$) from a Schwarzschild BH.
The associated value of the DM mass giving the full contribution to DM is denoted by $\bar m$.
We display the iso-contours of ${\rm Log}_{10} \bar m c^2$[GeV]  in the left panel of fig.\,\ref{fig-DM}, taken from \cite{Masina:2020xhk}.
The shaded area, which fully includes the region of BH domination ($\beta>\bar \beta$), is excluded by the constraints on structure formation, as we are going to discuss in the following.

Let us call $m_X$ the mass in the case of a Kerr BH with spin $a_*$, emitting the $X$ particle of spin $s_X=0,1/2,1,2$ (with $g_X=1,2,3,5$, respectively),
providing the full contribution to DM.
Using eq. (\ref{eq-NXphi}), the ratio $m_X/\bar m$ turns out to be, for radiation domination, in which case $Y_{BH}(t_{ev})$ is the same in the Kerr and Schwarzschild case,
\beq
\frac{m_X}{\bar m}
=\frac{N_0}{N_X}
= \frac{\tilde \phi_0(0)}{g_X \tilde \phi_{s_X}(a_*)} \,\, .
\label{eq-mXmbR}
\eeq
In the right panel of fig.\,\ref{fig-DM} we show the ratio $m_X/\bar m$ as a function of $a_*$. Notice that the Schwarzschild case for the various particle spins agrees with the findings of \cite{Auffinger:2020afu}.
In the more general Kerr case, we can see that the ratio $m_X/\bar m$ increases marginally as a function of $a_*$ for $s=0,1/2$, it slightly decreases for $s=1$, while drastically decreases for $s=2$.

For BH domination, using eq. (\ref{eq-YBHBH}) to take into account the difference in $Y_{BH}(t_{ev})$ for the Kerr and Schwarzschild case, one has
\beq
\frac{m_X}{\bar m}
= \alpha_K^{1/2} \frac{\tilde \phi_0(0)}{g_X \tilde \phi_{s_X}(a_*)} \,\, .
\label{eq-mXmbBH}
\eeq

\begin{figure}[h!]
\vskip .0 cm
 \begin{center}
 \includegraphics[width= 7.6 cm]{fig-DMl}    \,\,
\includegraphics[width= 7.6 cm]{fig-DM}
\end{center}
\caption{\baselineskip=15 pt \small  Left: Isocontorurs of ${\rm Log}_{10} \bar m c^2$[GeV] for a boson with $g_X=1$, in the Schwarzschild case,
from ref.\,\cite{Masina:2020xhk}. The shaded region, covering the region of BH domination and a small portion of radiation domination, is ruled out by constraints on structure formation.
Right: Ratio $m_X/\bar m$ as a function of $a_*$, for radiation domination, with $g_X=1,2,3,5$ and $s_X=0,1/2,1,2$ respectively. }
\label{fig-DM}
\vskip .2 cm
\end{figure}

\subsection{Constraints on warm DM}
 \label{sec-wdm}

If the $X$ particle is going to provide the full contribution to DM, one has to check that it was cold enough not to waste structure formation.
The $X$ particles are emitted with a distribution of momenta. An argument based on mean quantities \cite{Fujita:2014hha}
allows to derive a good estimate for the lower value of $m_X$ that would be compatible with structure formation.
We now generalize the argument of \cite{Fujita:2014hha, Masina:2020xhk, Baldes:2020nuv} to the Kerr case,
in order to inspect if the tension with structure formation is alleviated\footnote{A more sophisticated analysis
should follow the lines of \cite{Auffinger:2020afu}.}.

The momentum of the $X$ particle is red-shifted by the expansion of the Universe,
\beq
 p_{0} = \frac{a(t_{ev})}{a(t_{0})} p_{ev} = a(t_{EQ})  \frac{a(t_{ev})}{a(t_{EQ})} \frac{\langle E_X (t_{ev}) \rangle}{c}
 =  a(t_{EQ})  \frac{\langle E_X (t_{EQ}) \rangle}{c}  \,\, ,
\eeq
where we used eqs.\,(\ref{eq-meanE}), (\ref{eq-ETEQ}), $a(t_{0})=1$.
Assuming that it is no more relativistic, the velocity of the $X$ particle now is
\beq
 \frac{v_X}{c}= \frac{p_{0}}{c \,m_X}
 =  a(t_{EQ})\, \frac{\langle E_X (t_{EQ}) \rangle}{m_X c^2}
 =  {\alpha_K^{1/2}} \,  \frac { \tilde \delta_{s_X}} { \tilde \delta_{0}}\, \frac{\bar m }{m_X }   \frac{\bar v}{c}
 \label{eq-vX}
\eeq
where we used eqs.\,(\ref{eq-EmEQ}) and defined $\bar v/c$ to be the velocity of the previously introduced reference DM case,
that is a scalar (with $g_X=1$) in the Schwarzschild case, so that
\beq
\frac{\bar v}{c} =a(t_{EQ})\, \frac{1}{\bar m c^2} \,  \tilde \delta_{0} \left( \frac{M_{BH}}{1 \,{\rm g}} \right)^{1/2}\,0.11 \,{\rm keV}
= 0.9 \times 10^{-6}\, ,
\eeq
where in the last equation we used $a(t_{EQ})=\Omega_R/\Omega_{M} \approx 1.8\times 10^{-4}$.

For radiation domination, using eq.\,(\ref{eq-mXmbR}), eq. (\ref{eq-vX}) becomes
\beq
 \frac{v_X}{c}=g_X\, {\alpha_K^{1/2}}\,   \frac { \tilde x_{s_X}(a_*)} { \tilde x_{0}(0)}\,  \frac{\bar v}{c}\, ,
 \label{eq-vXR}
\eeq
while for BH domination, using eq.\,(\ref{eq-mXmbBH}), it becomes
\beq
 \frac{v_X}{c}=g_X\, \frac { \tilde x_{s_X}(a_*)} { \tilde x_{0}(0)}\,  \frac{\bar v}{c}\, ,
  \label{eq-vXBH}
\eeq

The velocity $v_X$ is subject to an upper bound that can be estimated to be comparable to the upper limit on the velocity of a warm thermal relic DM candidate.
Assume that a warm thermal relic giving the full contribution to DM was relativistic at decoupling. In order not to waste structure formation, the lower bound on its mass
is $m_{W} c^2 \gtrsim 3.5$ keV (at 2$\sigma$) \cite{Irsic:2017ixq}.
Since both the velocity and particle temperature scale as the inverse of the scale factor, and assuming entropy conservation from decoupling to the present epoch \cite{Bode:2000gq}
\footnote{If the warm particles decouple when relativistic, their momentum distribution function remains constant until gravitational clustering begins.
All particle momenta scale as $a^{-1}$ which we can describe by scaling their temperature $T_W$ accordingly.
When the particles become non-relativistic we can use $p = m_W v_W$.
}
\beq
 \frac{v_{W} }{c}
 \approx \frac{k_B T_W(t_{0})}{m_{W} c^2}
=\left( \frac{g_{*,S}(t_{r})}{g_{*,S}(t_{dec})}\right)^{1/3} \left(\frac{4}{11}\right)^{1/3} \frac{k_B T_{CMB}}{m_{W} c^2}\lesssim 0.7\times  10^{-8} \,\, ,
\label{eq-limV}
\eeq
where we used the fact that ${g_{*,S}(t_{r})}/{g_{*,S}(t_{dec})} = 11\, {\rm eV} /(m_W c^2)$ (see e.g. the discussion in \cite{Auffinger:2020afu}).

If thew $X$ particles are going to fully contribute to DM, we have to require $v_X\lesssim v_{W}$. Hence there is a tension with structure formation because $\bar v$ turns out to be bigger than $v_W$ by two orders of magnitude (consistently with the findings of \cite{Fujita:2014hha, Masina:2020xhk,Baldes:2020nuv,Auffinger:2020afu}). This rules out all the region of BH domination, and also a small confining portion of radiation domination, as can be seen from the left panel of fig.\,\ref{fig-DM}.

In the Schwarzschild case with non zero spins, the tension is reduced for increasing spin values, because of the suppression in $v_X$ coming from the factor $\tilde x_{s_X}(0)/\tilde x_0(0)$, see eqs.\,(\ref{eq-vXR}), (\ref{eq-vXBH}) and the right panel of fig.\,\ref{fig-tphi}.
But even for $s=2$ (with $g_X=5$) the tension persists because, as can be seen from fig.\,\ref{fig-tphi}, one has $ g_X \tilde x_2(0)/\tilde x_0{(0)} \approx 0.1$: the tension is thus reduced at the level of one order of magnitude, consistently with the findings of \cite{Auffinger:2020afu}.

In the more general Kerr case, we can derive from the right panel of fig.\,\ref{fig-tphi} what is the behavior of the quantity ${ \tilde x_{s_X}(a_*)}/ { \tilde x_{0}(0)}\, $ for increasing values of $a_*$.
For $s=0,1/2,1$ there is no significant difference with respect to the Schwarzschild case; for $s=2$,
since $\tilde x_{2}(a_*)$ is an increasing function of $a_*$, the tension with structure formation become definitely worse than in the Schwarzschild case.

This hows that  there is no possibility to save the BH domination scenario with Kerr BHs.

\section{Stable particles as dark radiation}

Let us assume that the $X$ particle is light enough to contribute to DR (in which case the contribution to DM is marginal).
The contribution of such a DR component to the effective number of relativistic dof is parametrized by
\beq
\Delta N_{eff} = \frac{\rho_{X}(t_{EQ})}{\rho_{R}(t_{EQ})}   \left(  N_\nu + \frac{8}{7} \left(  \frac{11}{4}\right)^{4/3} \right) \,,
\label{eq-defNeff}
\eeq
where $N_\nu=3.045$ \cite{deSalas:2016ztq}, and $t_{EQ}$ is the time of matter-radiation equality.

We now extend to the Kerr case the argument followed in ref.\,\cite{Hooper:2019gtx, Masina:2020xhk} to calculate $\Delta N_{eff}$.
The ratio of the energy density in DR with respect to radiation at matter-radiation equality is
\beq
\frac{\rho_{X}(t_{EQ})}{\rho_{R}(t_{EQ})}
=\frac{1} {{\alpha'}^{4/3}} \, \frac{\rho_{X}(t_{ev})}{\rho_{R}(t_{ev})}\,  \frac{   g_{*,S} (t_{EQ}) } {g_*(t_{EQ})   }  \frac{   g_{*,S} (t_{EQ})^{1/3} } {  g_{*,S} (t_{ev}) ^{1/3}  }\,.
\eeq

As shown e.g. in fig.\,3 of \cite{Auffinger:2020afu}, within the SM, $g_{*,S}(t_{ev})=106.75$ is constant for $M_{BH}<10^6$\,g
(while it drops down to the value $10.75$ at $M_{BH}\approx 10^{9}$ g).
Substituting the above expression in eq.\,(\ref{eq-defNeff}), and taking $g_{*,S} (t_{EQ})=3.94$, $g_{*} (t_{EQ})=3.38$, $N_\nu=3.045$, $g_{*,S} (t_{ev})=106.75$, one has
\beq
\Delta N_{eff}
\approx 2.89\,\frac{1}{ {\alpha'}^{4/3}} \, \frac{\rho_{X}(t_{ev})}{\rho_{R}(t_{ev})}  \,.
\eeq
In the BH mass range above $10^6$ g, there is a slight enhancement with respect to the factor $2.89$, which has to be substituted by $6.21$ for $M_{BH} = 10^{9}$ g.

The ratio $\rho_{X}(t_{ev})/ \rho_{R}(t_{ev})$ was already studied in eqs. (\ref{eq-rhoiR}), (\ref{eq-rhoiBH}).
For BH domination, within the SM with the additional DR, it was shown in the left panel of fig.\,\ref{fig-td}, for different particle spins.
For radiation domination the results have to be suppressed by $\beta/\bar\beta$.

For the Schwarzschild case and BH domination, the associated prediction for $\Delta N_{eff}$ is shown in the left panel of fig. \ref{fig-DR},
taking $\alpha'=1$, and for various particle spins, as indicated.
These results are in full agreement with \cite{Hooper:2019gtx}.
The present sensitivity to $\Delta N_{eff}$ of CMB observations is shown: since $N_{eff} = 2.99 \pm 0.17$\,\cite{Aghanim:2018eyx},
one has $ N_{eff}<3.33$ at $2\sigma$, or equivalently $\Delta N_{eff} =N_{eff}-N_\nu  <  0.29$ at $2\sigma$.
Interestingly enough, there are optimistic possibilities of detecting some signal in the future \cite{Hooper:2019gtx},
as the predicted contribution to $\Delta N_{eff}$ is potentially within the projected reach of stage IV experiments,
$\Delta N_{eff} \approx 0.02$: this is the case for DR particles with $s < 3/2$, but not for a massless or massive $s=2$ particle (called graviton for short).

The right panel of fig.\,\ref{fig-DR} shows the dependence of  $\Delta N_{eff}$ on the BH spin, $a_*$.
One can see that for  $s=0,1/2,1$ there are no significant changes, while for $s=2$ there is a significant increase.
However, for $s=2$, the contribution to $\Delta N_{eff}$ remains below the future observable region, unless
$a_*>0.9\, (0.75)$ in the massless (massive) case respectively, for $M_{BH}<10^6$\,g.
In the case of larger values of $M_{BH}$ (corresponding to evaporation just before nucleosynthesis),
there is a further slight increase, as the left panel of fig.\,\ref{fig-DR} shows.

\begin{figure}[h!]
\vskip .0 cm
 \begin{center}
 \includegraphics[width= 7.6 cm]{fig-DNeff}    \,\,\, \includegraphics[width= 7.6 cm]{fig-DNeffa}
\end{center}
\caption{\baselineskip=15 pt \small
Left: Various DR contributions to $\Delta N_{eff}$, as a function of the BH mass, assuming an epoch of BH domination and taking $\alpha'=1$.
Right: the dependence of  $\Delta N_{eff}$ on the BH spin $a_*$, for $M_{BH}<10^6$\,g.}
\label{fig-DR}
\vskip .2 cm
\end{figure}

We thus find less optimistic results than those presented in ref.\,\cite{Hooper:2020evu} for "hot" gravitons.
We comment here on the source of the discrepancy.
The latter work studies the case of BHs that are spinning as a consequence of having undergone
previous mergers. The distribution of angular momenta predicted for such a BH population peaks strongly at $a_*=0.7$,
almost entirely independently of the masses or the initial spin distribution of the merging binaries \cite{Fishbach:2017dwv}.
After integrating over the evolution of BHs with a distribution of initial spins as described in \cite{Fishbach:2017dwv},
ref.\,\cite{Hooper:2020evu} finds that approximately $f_G \approx 0.47\%$ of the energy emitted as Hawking radiation is in the form of massless "hot" gravitons. On the contrary, as appears from the left panel of fig.\,\ref{fig-td}, we find less optimistic values: in particular for a massless (massive) graviton and $a_*=0.7$, we have $f_G=\rho_G/\rho_R= 0.20\% (0.50\%)$; for an extremal BH with $a_*=0.97$, the latter values increase to $1.4\% (3.6\%)$ respectively.
Our significantly smaller estimate for $f_G$ is thus the source of the discrepancy in the associated "hot" graviton contribution to $\Delta N_{eff}$:
indeed, for ref.\,\cite{Hooper:2020evu}, $\Delta N_{eff}$ might be as large as  $0.01-0.03$ for $a_*=0.7$, and even as large as $\Delta N_{eff} =0.3$ for near extremal BHs. According to our findings, this would rather apply to the massive graviton case, but not to the massless one.

\section{Conclusions}

We have extended to the Kerr case the study of DM and DR from evaporating primordial BHs.

For DM, one might have expected \cite{Auffinger:2020afu} that the constraints from structure formation that exclude the scenario of BH domination for light DM in the Schwarzschild case, would have been softened. We find instead that, while for the lower spins as $s=0,1/2,1$ the tension is not changed significantly for all values of the BH spin $a_*$, for $s=2$ the tension is even enhanced with increasing values of $a_*$.
We conclude that invoking an angular momentum for the evaporating BHs, does not offer a solution to save the BH domination scenario.

A couple of possibilities to save BH domination should be mentioned.
As suggested in \cite{Fujita:2014hha}, some mechanism providing entropy non conservation and taking place after the evaporation of primordial BHs (like \emph{e.g.} moduli decay) might succeed this task.
In this work we considered non-interacting DM from primordial BHs evaporation, but allowing for self-interacting DM offers the possibility to escape the structure formation bound in the light case for BH domination \cite{Bernal:2020kse}: thermalization in the DM sector decreases the mean DM kinetic energy and, together with number-changing processes, can have a strong impact, in particular enhancing the DM relic abundance by several orders of magnitude.

For DR, it is well known \cite{Hooper:2019gtx, Masina:2020xhk} that in the Schwarzschild case particles with $s=0,1/2,1$ might give a contribution to
$\Delta N_{eff}$ at hand of future experimental sensitivity, while this does not applies to the massive or massless $s=2$ case (see the left panel of fig.\,\ref{fig-DR}).
In the Kerr case, we find that the contribution to $\Delta N_{eff}$ by particles with spin $s=0,1/2,1$ has a very mild dependence on $a_*$, while for $s=2$ the contribution significantly increases with $a_*$ \,\cite{Hooper:2020evu}.
In any case, for $s=2$ and a moderate value of the spin parameters like $a_*=0.7$, we find that $\Delta N_{eff}$ remains below the projected sensitivity in the massless case (while in the massive case it would reach the level of the planned sensitivity only for BHs evaporating just before nucleosynthesis),
see the right panel of fig.\,\ref{fig-DR}. In the massless case for $s=2$, only for extreme values of the spin parameter, $a_*>0.9$, $\Delta N_{eff}$ would reach the projected experimental sensitivity. Our results are thus less optimistic than those of ref.\,\cite{Hooper:2020evu} for the "hot" graviton case.

\section*{\Large Acknowledgements}

I.M. acknowledges partial support by the research project TAsP (Theoretical Astroparticle Physics) funded by the Istituto Nazionale di Fisica Nucleare (INFN).

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


These Deep Dish Meaty Pizza Muffins are easy to make because they are made with frozen yeast rolls.
Making individual pizza muffins is a great way for everyone to get the toppings they enjoy on their pizza.
Our family looks forward to Friday nights and spending time together. Making an effort to spend time together is more difficult now my kids are young adults and teenagers but just as important. They have extremely busy schedules, and it can be hard to get everyone together. My oldest daughter is in her third year of medical school studying to be a doctor. My oldest boys are twins. One is in college, and the other one is home with me, Caleb, my special needs son. I also have a seventeen-year-old son and a thirteen-year-old daughter.
These Deep Dish Meaty Pizza Muffins are super easy because they are made with frozen yeast rolls. Making individual pizza muffins are an easy way for everyone to get the toppings they enjoy on their pizza.
We have always tried to make Friday nights pizza and game night. Now that my kids are older, we play cards on Friday nights after pizza. It is a great way to spend time with my family learning about their week.
I often try different types of pizza recipes for our family pizza night. I have made deep dish skillet pizzas, slow cooker pizzas, tortilla pizzas, and more. My family seems to enjoy these pizza muffins because they can make their pizza the way they like it. I used frozen yeast rolls to make these pizza muffins.
The frozen yeast dough tastes more like homemade pizza dough.
The bottoms and sides of these pizza muffins remind me of Pizza Hut deep dish pizza. They are delicious with a buttery and crispy crust. These pizza muffins are easy to make, but they do require time for the dough to rise. If you enjoy a thick crust that is buttery and crispy, you will love these pizza muffins.
How to make muffin pizzas?
I thawed out twelve Rhodes frozen dinner rolls. One in each cupcake tin. After the rolls begin to thaw in about 30 minutes, I cut them into four equal pieces and placed the rolls back into the pan to rise. After they double in size, about three hours, brush six tablespoons of melted butter evenly all over the rolls.
I added our favorite pizza toppings for a meat lovers pizza in this recipe. Sometimes, I make a pizza topping bar where everyone can make their own. I used a tablespoon of each cooked ground sausage, cooked crumbled bacon, and chopped ham. To make it gooey and tasty, I topped these pizza muffins off with a tablespoon pizza sauce, and cheese. Before placing the pizza in the oven, I season it well with Paul Prudhomme's Pizza Pasta Magic. This seasoning is my favorite Italian seasoning. It will make even a frozen pizza seem incredibly delicious.
12 tablespoons each cooked ground sausage crumbled bacon, and chopped ham. One tablespoon of each per muffin tin.
12 tablespoons pizza sauce one tablespoon for each muffin tin.
12 tablespoons grated cheese one tablespoon for each muffin tin.
Lightly spray the muffin tins with a cooking spray.
Add one frozen dinner roll to each muffin tin.
After they begin to thaw, about 30 minutes, cut the dinner rolls into four equal parts. I cut them lengthwise and then across.
Allow the rolls to double in size about three hours.
Brush the melted butter equally on the twelve rolls.
Add one tablespoon of each sausage, bacon, ham, pasta sauce, and cheese.
Top with a pizza seasoning like Paul Prudhomme's Pizza Pasta Magic.
Bake 375 degrees for about 22 minutes.
Makes 12 cupcake size pizza muffins.
Diane, these look amazing! We love pizzas around here and these would be perfect for the grand kids to make their own way, with the toppings they like best. GREAT post and thanks for sharing!!!


A slightly different experiment here – I tried to use the rather powerful capabilities hidden within 3dsMax itself – Nitrous. By switching to the Quicksilver Renderer, and then making use of the NPR Explorer (which gives hidden, custom, fine tuned options on how the graphical styles turn out), it is then possible to create some very interesting images from a resource that is usually overlooked.
One of the added benefits with Nitrous and Quicksilver is that standard lighting options can still be used – so I added a daylight sun and then modified the lighting until I was content with the fall of the shadow and glare intensity.
The graphical style used for this image was "Graphite", and then the aforementioned NPR Explorer was used to modify the line weight, black/white levels, and even the angle of the pencil strokes.
Turned out quite well, and the image rendered under two minutes. Too bad that I canned the architectural idea I was pursuing with this image and a few others. Still – it probably would work well for a desktop background.